I live in a duplex and try to be as respectful as I can with my music volume. My new neighbor complained about the volume of my music. So I've been careful about it. Lo and behold, they're complaining again. There's no way I'm making that much noise. They keep threatening to get our property manager involved. What's a good tool/way to measure noise levels?

Putting up an additional layer of drywall can help dramatically, and it's not very difficult.

A layer of 5/8's adhered (16" on center) with a soft adhesive or "green glue" can work wonders.
Don't forget to remove the few screws that held the sheets up once the glue dries...you want the new drywall layer decoupled from the old wall, and leaving the screws can help create a sonic "short circuit" to the original wall.

Better yet is to frame a wall directly across (but not touching) your shared (common) wall, using flimsy 1-5/8ths metal studs or resilient channel, 2' on center, with 5/8ths sheathing.

This flimsy wall (aside from acting as a large panel bass absorber) will almost completely decouple your room from theirs, except for the attachments at the ceiling and floor.
These points can also be somewhat decoupled using rubber strips (I use cheap 10' lengths of rubber, garage door bottom seal from the depot) above and below your ceiling and floor starter tracks.

In the past, all common/shared walls where always built out of solid masonry, to avoid the possibility of noise disturbance from the adjacent living space, and also to act as a fire stop.

Soundproofing is the best way out but just out of curiosity, do they live so close by? They are neighbors who will always start trouble irrespective of whether or not there is a genuine reason to do so.

Soundproofing is the best way out but just out of curiosity, do they live so close by? They are neighbors who will always start trouble irrespective of whether or not there is a genuine reason to do so.

Based on what ABtoJayZ, the house they are living in is a duplex, meaning it is a house divided in two. Sounds travel faster through solids, so it is no surprise if the neighbors will complain because the barrier won't block the sound.